Orlicz Space on Groupoids

Let $G$ be a locally compact second countable groupoid with a fixed Haar system $\lambda =\{{\lambda }^u{\}}_{u\in G^0}$ and $(\mathrm{\Phi },\mathrm{\Psi })$ be a complementary pair of $N$-functions satisfying ${\mathrm{\Delta }}_2$-condition. In this article, we introduce the continuous field of Orlicz space $(L_0^{\mathrm{\Phi }},{\mathrm{\Delta }}_1)$ and provide a sufficient condition for the space of continuous sections vanishing at infinity, denoted $E_0^{\mathrm{\Phi }}$, to be an Banach algebra under a suitable convolution. Further, the condition for a closed $C_b(G^0)$-submodule $I$ of $E_0^{\mathrm{\Phi }}$ to be a left ideal is established. Moreover, we provide a groupoid analogue of the characterization of the space of convolutors of Morse-Transue space for locally compact groups.